Topology MOC

Topology counterexamples MOC

The following is a list of rather pathological topological spaces that serve as nice counterexamples.

Pathological topologies on arbitrary sets

• Trivial topology is not Hausdorff
• Discrete topology is Completely disconnected

Pathological manifolds

• Line with two origins is path-connected but not arc-connected (and not Hausdorff)
• Long line

Miscellaneous

• Topologist's sine curve is connected but not path-connected

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